Exercise 2, § 8

Let X be a compact topological space, R a separated equivalence relation in X and B a closed subalgebra of $\mathscr{C}(X)$ containing $A_R$ (notation of Exercise 1). If $f\in \mathscr{C}(X)$ coincides on each class $c$ following R with an element $g_c$ of B, then $f\in B$. (Let $\varepsilon >0$. There exists a saturated open neighbourhood $V_c$ of $c$ such that $|f-g_c|\leqslant \varepsilon$ on $V_c$. Let $V_{c_1}, . . . ,V_{c_n}$ cover X. Let $(u_1, . . . , u_n)$ be a partition of unity subordinate to $(V_{c_1}, . . . ,V_{c_n})$ and formed of functions of $A_R$. Then

and $|f-g|\leqslant \varepsilon .)$